L(s) = 1 | + (0.909 − 0.415i)2-s + (−0.0359 − 0.122i)3-s + (0.654 − 0.755i)4-s + (−1.78 − 1.34i)5-s + (−0.0836 − 0.0965i)6-s + (0.277 − 0.0399i)7-s + (0.281 − 0.959i)8-s + (2.51 − 1.61i)9-s + (−2.18 − 0.487i)10-s + (2.07 − 4.53i)11-s + (−0.116 − 0.0530i)12-s + (3.69 + 0.531i)13-s + (0.236 − 0.151i)14-s + (−0.101 + 0.267i)15-s + (−0.142 − 0.989i)16-s + (−5.80 + 5.02i)17-s + ⋯ |
L(s) = 1 | + (0.643 − 0.293i)2-s + (−0.0207 − 0.0707i)3-s + (0.327 − 0.377i)4-s + (−0.797 − 0.603i)5-s + (−0.0341 − 0.0394i)6-s + (0.105 − 0.0151i)7-s + (0.0996 − 0.339i)8-s + (0.836 − 0.537i)9-s + (−0.690 − 0.154i)10-s + (0.624 − 1.36i)11-s + (−0.0335 − 0.0153i)12-s + (1.02 + 0.147i)13-s + (0.0631 − 0.0405i)14-s + (−0.0261 + 0.0689i)15-s + (−0.0355 − 0.247i)16-s + (−1.40 + 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.407 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.407 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36834 - 0.888248i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36834 - 0.888248i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.909 + 0.415i)T \) |
| 5 | \( 1 + (1.78 + 1.34i)T \) |
| 23 | \( 1 + (0.919 - 4.70i)T \) |
good | 3 | \( 1 + (0.0359 + 0.122i)T + (-2.52 + 1.62i)T^{2} \) |
| 7 | \( 1 + (-0.277 + 0.0399i)T + (6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (-2.07 + 4.53i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-3.69 - 0.531i)T + (12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (5.80 - 5.02i)T + (2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (2.73 - 3.15i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (1.53 + 1.77i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-3.18 - 0.936i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-4.55 - 7.09i)T + (-15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (-1.35 - 0.868i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (2.28 + 7.78i)T + (-36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + 5.98iT - 47T^{2} \) |
| 53 | \( 1 + (-10.5 + 1.51i)T + (50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (1.46 - 10.1i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-10.4 - 3.05i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (8.30 - 3.79i)T + (43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (1.57 + 3.44i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-7.70 - 6.67i)T + (10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-0.687 + 4.77i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (6.83 + 10.6i)T + (-34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (5.36 - 1.57i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (2.46 - 3.83i)T + (-40.2 - 88.2i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.93112007033593357200863227273, −11.37890752604572110702005176749, −10.37915441723146379358028168857, −8.922723458670411916463504011281, −8.241838374225183994449037747223, −6.71214602574709265366387273175, −5.85083176861139730790755506786, −4.17241738502015352792848156834, −3.71357402128797328941023911053, −1.36411689778273897670276368372,
2.40510781405899034729449825144, 4.14046586755876462404530840112, 4.67587474936546925386261968613, 6.58143772034018478503301604613, 7.07808349487622073923528940306, 8.192624414450432862030936815321, 9.480450834273447992019772949056, 10.81551797187430408622482625592, 11.37456687995156333271235549001, 12.52264800209957691999341673910